If the time scale for cross-sectional mixing is comparable with or larger than the flow period, then, after each flow reversal, there can be a substantial time span in which the contaminant cloud is contracting. Thus, the apparent longitudinal-diffusion coefficient is negative. This means that the contaminant dispersion cannot be modelled by a diffusion equation, because negative diffusivities imply the spontaneous development of infinite concentrations. Here it is shown how this periodic contracting and expanding can be modelled by a delay-diffusion equation (Smith 1981) \[ \partial_t\overline{c} + \overline{u}\partial_x\overline{c} = \overline{\kappa}\partial^2_x\overline{c} + \int_0^{\infty} \partial_{\tau}D\partial^2_x\overline{c}(x-X,t-\tau)\,d\tau, \] where $\overline{u}(t)$ is the bulk velocity, X(t, τ) a coordinate displacement, and D(t, τ) the diffusion coefficient at time τ after discharge. The recent memory ∂τD is always positive and diffusive in character, so singularities cannot arise. However, when τ is large this memory function can be negative because of reversed flow at earlier times. Particular attention is given to estuarial flows and results are derived for the dependence of D upon the water depth and upon the width of the estuary.

A numerical method is developed for computing steady supercritical flow about an ellipse at zero angle of attack. The flow is assumed to be two-dimensional, inviscid, isentropic and irrotational. The free-stream Mach number lies in the high subsonic range so that a shock wave occurs locally near the body. The full potential equations are solved by Telenin's method and the ‘method of lines’. Smooth interpolating functions are assumed for the unknown flow variables in selected co-ordinate directions. The resulting set of ordinary differential equations is then integrated away from or along the body depending upon whether the flow is smooth or discontinuous. Jump conditions of the governing equations are applied across the shock wave so that it is perfectly sharp. A doublet solution for flow past a closed body is used as the farfield boundary condition. Supercritical flow calculations have been performed for ellipses with thickness ratio of 0·2 and 0·4 at various free-stream Mach numbers. The present results are compared with the shock-capturing method, and good agreement is obtained.

The techniques of uniform-slender-body theory are employed to investigate the hydrodynamic forces and moments acting on a moving ship in shallow water and the interaction forces between two such ships on parallel courses. Of particular interest is the verification by these methods of the validity of the solutions by matched asymptotic expansions constructed by previous authors. The free surface is assumed rigid and each ship is modelled as a slender body of revolution located midway between two closely spaced parallel planes. The velocity potential due to the presence of a single ship is represented as the potential due to singularities distributed along a portion of the axis inside the body, together with appropriate image singularities outside the body. The boundary condition on the body leads to a linear integral equation for the density of singularities, which is solved using the asymptotic analysis discussed by Geer (1975). The sinkage force and trimming moment on the vessel are computed. When two ships are moving on parallel courses, appropriate interaction potentials are introduced in a manner similar to that for a single ship and the integral equations resulting from the application of the boundary condition are solved asymptotically. The interaction forces and moments between the ships are computed and compared with some experimental and other theoretical results.

Two of the simpler nonlinear wave systems on water of uniform depth are permanent waves and wave groups of permanent envelope. The interaction of each of these wave systems with swell of much smaller amplitude and greater wavelength, propagating in the same direction, is investigated analytically and numerically. A linear-stability analysis of the modulation of these systems by swell shows that they are unstable over short times. Calculations of their evolution over longer times confirms that the initial exponential growth of the modulations is not sustained, and that cyclic recurrence of the modulations occurs in some cases. The modulation of a wave train by swell is found to concentrate the energy of the wave train into single waves in turn, a process which may cause irreversible nonlinear changes such as wave breaking. In contrast, the only observable effect in the modulation of a wave group by swell is a small slow oscillation of the envelope of the group as it propagates. The conclusion is that wave trains on the ocean, generated for example by a wind system of long fetch and duration, disintegrate under the modulation of swell. Wave groups, however, either wind-generated or resulting from the breakdown of wave trains, propagate almost unchanged by the presence of swell.

Finite-amplitude convection rolls of an infinite-Prandtl-number fluid in a long channel heated from below are investigated. Because of the side walls, the convection rolls depend on all three spatial co-ordinates, although only two velocity components are of importance for a wide range of Rayleigh numbers and aspect ratios. Accurately converged solutions are presented for a range of aspect ratios between 0 (Bénard convection) to 100 (Hele Shaw convection) and for Rayleigh numbers up to about 50 times the critical linear stability value. The influence of rigid versus slip boundaries as well as the wavelength of the convection rolls on the heat transport is investigated in detail. Comparisons with existing results for the analogous problem of convection in a porous medium indicates that the similarity tends to disappear at Rayleigh numbers less than a few times the critical value. Whenever possible, the theoretical findings are compared with experimental results. In all cases close agreement is found.

Some general results are derived for the efficiency of energy absorption of a system of uniform oscillatory surface pressure distributions. The results, which are based on classical linear water-wave theory, show the close analogies which exist with theories for systems of absorbing oscillatory rigid bodies and a number of new reciprocal relations for pressure distributions are suggested and proved. Some simple examples illustrating the general results are given and compared with the corresponding results for rigid bodies.

Effects of screens and perforated plates (grids) on free-stream turbulence are studied in several test flow conditions. The level, structure and decay of the turbulence generated by such ‘manipulators’ depend in part on their shear-layer instabilities, and can therefore be modified by inserting additional devices immediately downstream. The performance of screens and some perforated plates is found to depend on the characteristics of the incoming flow such as velocity, turbulence level and spectra. Combinations of perforated plates and screens are found to be very effective flow manipulators. By optimizing the intermanipulator separation and carefully matching the scales between the manipulator pair, the turbulence decay rate downstream of a grid can be quadrupled.